Olympiad problems

2005 Paraguay Mathematical Olympiad, 4

Tags: number theory

In the expression $t=\frac{8a+ 1}{b}$ where $a, b, t$ are positive integers, where $b <7$. Determine the values of $a$ and$ b$ that allow to obtain $t$ under the established conditions.

2008 Pan African, 2

Tags: geometry

Let $C_1$ be a circle with centre $O$, and let $AB$ be a chord of the circle that is not a diameter. $M$ is the midpoint of $AB$. Consider a point $T$ on the circle $C_2$ with diameter $OM$. The tangent to $C_2$ at the point $T$ intersects $C_1$ at two points. Let $P$ be one of these points. Show that $PA^2+PB^2=4PT^2$.

2013 Iran Team Selection Test, 16

Tags: function , functional equation , algebra

The function $f:\mathbb Z \to \mathbb Z$ has the property that for all integers $m$ and $n$ \[f(m)+f(n)+f(f(m^2+n^2))=1.\] We know that integers $a$ and $b$ exist such that $f(a)-f(b)=3$. Prove that integers $c$ and $d$ can be found such that $f(c)-f(d)=1$. [i]Proposed by Amirhossein Gorzi[/i]

1974 Chisinau City MO, 79

Tags: combinatorics , sum

There are many of the same regular triangles. At the vertices of each of them, the numbers $1, 2, 3$ are written in random order. The triangles were superimposed on one another and found the sum of the numbers that fell into each of the three corners of the stack. Could it be that in each corner the sum is equal to: a) $25$, b) $50$?

2004 Iran MO (3rd Round), 20

Tags: algebra , polynomial , number theory

$ p(x)$ is a polynomial in $ \mathbb{Z}[x]$ such that for each $ m,n\in \mathbb{N}$ there is an integer $ a$ such that $ n\mid p(a^m)$. Prove that $0$ or $1$ is a root of $ p(x)$.

2024 Poland - Second Round, 1

Tags: algebra

Does there exist a rational $x_1$, such that all members of the sequence $x_1, x_2, \ldots, x_{2024}$ defined by $x_{n+1}=x_n+\sqrt{x_n^2-1}$ for $n=1, 2, \ldots, 2023$ are greater than $1$ and rational?

2006 Stanford Mathematics Tournament, 11

Tags: probability

An insurance company believes that people can be divided into 2 classes: those who are accident prone and those who are not. Their statistics show that an accident prone person will have an accident in a yearly period with probability 0.4, whereas this probability is 0.2 for the other kind. Given that 30% of people are accident prone, what is the probability that a new policyholder will have an accident within a year of purchasing a policy?

2011 USAMO, 4

Tags: modular arithmetic

Consider the assertion that for each positive integer $n\geq2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of $4$. Either prove the assertion or find (with proof) a counterexample.

2011 AMC 12/AHSME, 18

Tags: parabola , function , absolute value

Suppose that $|x+y|+|x-y|=2$. What is the maximum possible value of $x^2-6x+y^2$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9 $

2008 AMC 12/AHSME, 17

Tags: parabola , geometry , analytic geometry , calculus , graphing lines , slope , conic

Let $ A$, $ B$, and $ C$ be three distinct points on the graph of $ y\equal{}x^2$ such that line $ AB$ is parallel to the $ x$-axis and $ \triangle{ABC}$ is a right triangle with area $ 2008$. What is the sum of the digits of the $ y$-coordinate of $ C$? $ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 20$

2023 ISL, C7

Tags: combinatorics , graph theory

The Imomi archipelago consists of $n\geq 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the traveller starts at, to visit all the islands exactly once (in particular, not returning to the island the traveller started at). Determine the maximal possible value of $k$ in terms of $n$. [i]Anton Trygub, Ukraine[/i]

2005 Miklós Schweitzer, 2

Tags: number theory with sequences , integer sequence , periodic function , irrational number , ceiling function , number theory

Let $(a_{n})_{n \ge 1}$ be a sequence of integers satisfying the inequality \[ 0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1 \] for all $n \ge 2$. Prove that the sequence $(a_{n})$ is periodic. Any Hints or Sols for this hard problem?? :help:

2006 Austrian-Polish Competition, 2

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients satisfying the equation \[(x+1)^{3}P(x-1)-(x-1)^{3}P(x+1)=4(x^{2}-1) P(x)\] for all real numbers $x$.

2008 Romania Team Selection Test, 3

Tags: circumcircle , geometry

Let $ ABCDEF$ be a convex hexagon with all the sides of length 1. Prove that one of the radii of the circumcircles of triangles $ ACE$ or $ BDF$ is at least 1.

1985 Traian Lălescu, 2.2

Tags: algebra

Let $ a,b,c\in\mathbb{R} , E=(a-b)^2(b-c)^2(c-a)^2, $ and $ S_k=a^k+b^k+c^k,\forall k\in\{ 1,2,3,4\} . $ Write $ E $ in terms of $ S_k. $

2021 Taiwan TST Round 2, G

Tags: geometry , circumcircle , geometric transformation , reflection

Let $ABC$ be a triangle with circumcircle $\Gamma$, and points $E$ and $F$ are chosen from sides $CA$, $AB$, respectively. Let the circumcircle of triangle $AEF$ and $\Gamma$ intersect again at point $X$. Let the circumcircles of triangle $ABE$ and $ACF$ intersect again at point $K$. Line $AK$ intersect with $\Gamma$ again at point $M$ other than $A$, and $N$ be the reflection point of $M$ with respect to line $BC$. Let $XN$ intersect with $\Gamma$ again at point $S$ other that $X$. Prove that $SM$ is parallel to $BC$. [i] Proposed by Ming Hsiao[/i]

2001 China Team Selection Test, 2

Tags: trigonometry , inequality , sequence , inequalities , analysis

Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds: $\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$

2005 Today's Calculation Of Integral, 89

Tags: calculus , integration , trigonometry , function , symmetry , calculus computations

For $f(x)=x^4+|x|,$ let $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$ Find the value of $\frac{I_1}{I_2}.$

2013 China Team Selection Test, 3

Tags: analytic geometry , combinatorics , geometry

A point $(x,y)$ is a [i]lattice point[/i] if $x,y\in\Bbb Z$. Let $E=\{(x,y):x,y\in\Bbb Z\}$. In the coordinate plane, $P$ and $Q$ are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let $T=P\cap Q$. Prove that if $T\ne\emptyset$ and $T\cap E=\emptyset$, then $T$ is a non-degenerate convex quadrilateral region.

2000 Iran MO (3rd Round), 1

Tags: geometry

Let us denote $\prod = \{(x, y) | y > 0\}$. We call a [i]semicircle[/i] in $\prod$ with center on the $x-\text{axis}$ a [i]semi-line[/i]. Two intersecting [i]semi-lines [/i]determine four [i]semi-angles[/i]. A bisector of a [i]semi-angle [/i]is a [i]semi-line [/i]that bisects the [i]semi-angle[/i]. Prove that in every [i]semi-triangle [/i](determined by three [i]semi-lines[/i]) the bisectors are concurrent.

2016 Bosnia and Herzegovina Junior BMO TST, 3

Tags: geometry

Let $O$ be a center of circle which passes through vertices of quadrilateral $ABCD$, which has perpendicular diagonals. Prove that sum of distances of point $O$ to sides of quadrilateral $ABCD$ is equal to half of perimeter of $ABCD$.

2007 IMO Shortlist, 6

Tags: geometry , circumcircle , symmetry , ratio

Determine the smallest positive real number $ k$ with the following property. Let $ ABCD$ be a convex quadrilateral, and let points $ A_1$, $ B_1$, $ C_1$, and $ D_1$ lie on sides $ AB$, $ BC$, $ CD$, and $ DA$, respectively. Consider the areas of triangles $ AA_1D_1$, $ BB_1A_1$, $ CC_1B_1$ and $ DD_1C_1$; let $ S$ be the sum of the two smallest ones, and let $ S_1$ be the area of quadrilateral $ A_1B_1C_1D_1$. Then we always have $ kS_1\ge S$. [i]Author: Zuming Feng and Oleg Golberg, USA[/i]

1997 All-Russian Olympiad, 1

Tags: quadratic , modular arithmetic , polynomial , algebra

Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots? [i]N. Agakhanov[/i]

Denmark (Mohr) - geometry, 1993.4

Tags: trisector , area , geometry

In triangle $ABC$, points $D, E$, and $F$ intersect one-third of the respective sides. Show that the sum of the areas of the three gray triangles is equal to the area of middle triangle. [img]https://1.bp.blogspot.com/-KWrhwMHXfDk/XzcIkhWnK5I/AAAAAAAAMYk/Tj6-PnvTy9ksHgke8cDlAjsj2u421Xa9QCLcBGAsYHQ/s0/1993%2BMohr%2Bp4.png[/img]

2004 China Western Mathematical Olympiad, 1

Tags: number theory

Find all integers $n$, such that the following number is a perfect square \[N= n^4 + 6n^3 + 11n^2 +3n+31. \]